Encyclopedia of Microtonal Music Theory

144-tone equal-temperament / 144-edo

The tuning determined by 144 logarithmically equal divisions of the "octave". (see also EDO)

144-edo has been advocated by Dan Stearns and Joe Monzo for notational purposes, as an easily-grasped representation of the virtual pitch continuum. Their adaptation of "accidental" symbols is based on the 72-edo notation created by Ezra Sims, and on Monzo's ASCII adaptation of that, called HEWM.

Below is a table of the range of the virtual pitch continuum which can be represented by each 144-edo note. Only one semitone is shown, as the data can be reproduced similarly for the other 11 semitones of 12-edo. The table gives the absolute Semitone ranges (i.e., cents as a fraction of semitones), the nearest 12-edo degree and its cents deviation (i.e., to relate 144-edo to Johnny Reinhard's 1200-edo notation), and MIDI pitch-bend values (in 12mus) for the deviation from 12-edo.

This emphasizes the use of 144-edo as a notation, rather than as an actual tuning. (The latter would give the precise cent or pitch-bend value of each 144-edo degree, which I give further below.) This is how Stearns and Monzo use 144-edo to represent pitches which they expect to actually be tuned slightly differently, usually just-intonation in Monzo's case, and poly-edos in Stearn's case. (An example of Monzo's use of 144-edo to represent just-intonation pitches which fall within the ranges given below, can be found in his score to A Noiseless Patient Spider.)

144-edo is calculated by taking the 144th root of each successive power of 2, from 0 to 143, with higher or lower "octaves" of these 144 notes assumed to be, and tuned as, equivalents. The basic step-size is 8 & 1/3 cents.

Below is an overview of the complete 144-edo scale as a tuning. Notes which are presented horizontally adjacent are enharmonically equivalent.

Note also that in addition to the regular 12edo "5th" of 700 cents, 144edo also provides a quasi-meantone-type "5th" of ~6912/3 cents. This is very close to the "5th" sizes of 33edo (= ~69010/11 cents) and 1/2-comma meantone (= ~6911/5 cents), which are the bottom limit for meantone-like systems (because the note functioning as a "tone", the ii = 2nd degree of the diatonic scale, is tempered a full comma flat so that it is already exactly or nearly exactly the ratio 10/9 instead of 9/8, and thus leaves no room for a "mean" tone).

Note also that 144-edo provides a very likely explanation for the tetrachord divisions of the ancient Greek music-theorist Aristoxenus.

[Joe Monzo]

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